qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
3,270,725 | <p>Hello everyone I read on my notes this proposition: </p>
<p>Given a field <span class="math-container">$K$</span> and <span class="math-container">$R=K[T]$</span>, let <span class="math-container">$M$</span> be a (left) finitely generated <span class="math-container">$R$</span>-module; then <span class="math-contai... | hmakholm left over Monica | 14,366 | <p>Instead of dividing according to the sign of <span class="math-container">$x$</span>, give <span class="math-container">$\tan x$</span> a name such as <span class="math-container">$y$</span>, and ask yourself:</p>
<p>For which <span class="math-container">$y$</span> do we have <span class="math-container">$|y|>y... |
2,009,557 | <p>I am pretty sure this question has something to do with the Least Common Multiple. </p>
<ul>
<li>I was thinking that the proof was that every number either is or isn't a multiple of $3, 5$, and $8\left(3 + 5\right)$.</li>
<li>If it isn't a multiple of $3,5$, or $8$, great. You have nothing to prove.</li>
<li>But if... | Amr | 434,994 | <ul>
<li>proof by strong induction for n,a,b.</li>
<li>proof hypotheses p(n) = 3*a + 5*b for all n >= 8 //any number greater than 7 consists of 3s and 5s.</li>
<li>base case p(8) holds because it consists of 3 and 5.</li>
<li>Inductive step: we assume p(n) holds for all n>=8 we must prove that p(n+1) holds: <br>
since ... |
3,764,030 | <p>If <span class="math-container">$|z| = \max \{|z-1|,|z+1|\}$</span>, then:</p>
<ol>
<li><span class="math-container">$\left| z + \overline{z} \right| =1/2$</span></li>
<li><span class="math-container">$z + \overline{z} =1$</span></li>
<li><span class="math-container">$\left| z + \overline{z} \right| =1$</span></li>
... | Khosrotash | 104,171 | <p>Welcome to MSE. note that <span class="math-container">$|Z|,|z-1|,|z+1|$</span> are real numbers, so we can solve the equation, here split in two cases
<span class="math-container">$$ \max \{|z-1|,|z+1|\}=|z-1| or |z+1|\\
(1):|z| =|z-1|\to put \space z=x+iy\\|x+iy|=|x+iy-1|\\\sqrt{x^2+y^2}=\sqrt{(x-1)^2+y^2}\\x^2+y^... |
2,635,635 | <p>I have searched a lot and don't really understand the answers I've come across. So apologies in advance if I'm repeating a common question.</p>
<p>The problem is as follows: Distribute $69$ identical items across
$4$ groups where each groups needs to contain at least $5$ items.</p>
<p>The way I see the problem: $... | Rohan Shinde | 463,895 | <p>You are completely right.</p>
<p>Distribute 5 items to each $x_i$ before using star and bars.
Hence we get to find non negative integral solutions of $$a+b+c+d=49$$
Hence the answer would be $\binom {52}{3}$</p>
|
1,353,015 | <p>Given a positive singular measure $\mu$ on $[-\pi,\pi]$, we define a singular inner function by</p>
<p>$$S(z)=\exp\left(-\int_{-\pi}^{\pi}\frac{e^{i\theta}+z}{e^{i\theta}-z}\,d\mu(\theta)\right).$$</p>
<p>It is stated in many different sources that the radial limits $\lim_{r\rightarrow1^{-}}S(re^{2\pi it})$ equal ... | zhw. | 228,045 | <p>Hint: For $\mu$-a.e. $x\in \text {supp} (\mu),$</p>
<p>$$\lim_{r\to 0^+}\frac{1}{2r}\int_{(x-r,x+r)}d\mu = \infty.$$</p>
<p>Show this implies that $P_r(\mu)(x) \to \infty,$ where $P_r$ is the Poisson kernel. Now what is the relationship between $|S(re^{ix})|$ and $P_r(\mu)(x)?$</p>
|
2,867,521 | <p>I am interested in calculating the following double summation:</p>
<p>$\sum_{n=2}^ \infty \sum_{k =0}^{n-2}\frac{1}{4}^k \frac{1}{2}^{n-k-2}$</p>
<p>I don't really know where to start, so I was hoping someone could point me to some resource where I could learn the terminology/methodology associated with solving su... | Andreas Blass | 48,510 | <p>No. Let the directed set $\Lambda$ consist of the natural numbers, ordered as usual, and infinitely many additional elements, incomparable with each other but each $<$ all of the natural numbers. Then a net indexed by $\Lambda$ converges iff the sequence of its values at the natural numbers converges, while the v... |
2,867,521 | <p>I am interested in calculating the following double summation:</p>
<p>$\sum_{n=2}^ \infty \sum_{k =0}^{n-2}\frac{1}{4}^k \frac{1}{2}^{n-k-2}$</p>
<p>I don't really know where to start, so I was hoping someone could point me to some resource where I could learn the terminology/methodology associated with solving su... | Daniel Schepler | 337,888 | <p>Suppose the directed set you use as indices is $\mathbb{Z}$, and the net $\mathbb{Z} \to \mathbb{R}$ sends $n \mapsto 2^{-n}$. Then the net converges to 0; however, the image of the net along with its limit is $\{ 2^n \mid n \in \mathbb{Z} \} \cup \{ 0 \}$ which is unbounded so it cannot be compact or even relative... |
2,193,779 | <p><a href="https://i.stack.imgur.com/d65g2.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/d65g2.png" alt="enter image description here"></a></p>
<p>Any idea where the missing 3^k+2 comes from?
(sorry for the format, this thing didn't allow me to post images)</p>
| LM2357 | 376,702 | <p>The last step is simply
$$(k+1)(3^{k+1})+\frac{(2k-1)(3^{k+1})+3}{4}$$
$$\frac{4(k+1)(3^{k+1})+{(2k-1)(3^{k+1})+3}}{4}$$
$$\frac{(3^{k+1})(4k+4+2k-1)+3}{4}$$
$$\frac{(3^{k+2})(2k+1)+3}{4}$$
They have simply taken the $3$ out
$$3\left(\frac{(3^{k+1})(2k-1)+1}{4}\right)$$</p>
|
362,250 | <p>Let <span class="math-container">$\nu$</span> be a <em>finite</em> Borel measure on <span class="math-container">$\mathbb{R}^n$</span> and define the shift operator <span class="math-container">$T_a$</span> on <span class="math-container">$L^p_{\nu}(\mathbb{R}^n)$</span> by <span class="math-container">$f\to f(x+a)$... | Nik Weaver | 23,141 | <p>Well, for large <span class="math-container">$a$</span> the norm goes to infinity. Find a ball <span class="math-container">$B$</span> such that <span class="math-container">$\nu(B) > \nu(\mathbb{R}^n) - \epsilon$</span> and consider the characteristic function of <span class="math-container">$B$</span> shifted b... |
3,829,894 | <p>The problem is to find the parametric equation of the line that is tangent to the line of intersection between the plane <span class="math-container">$x+2y+3z=6$</span> and the surface <span class="math-container">$x^2+y^2=2$</span> and passes through the point <span class="math-container">$(1,1,1)$</span>.</p>
<p>H... | gt6989b | 16,192 | <p>Why not reduce to
<span class="math-container">$$
\int_0^1 x^m (1-x)^n dx
$$</span>
and take it by parts, differentiating <span class="math-container">$(1-x)^n$</span> until it disappears?</p>
<p>E.g. if <span class="math-container">$u = (1-x)^n$</span> and <span class="math-container">$dv = x^m dx$</span> then <spa... |
3,829,894 | <p>The problem is to find the parametric equation of the line that is tangent to the line of intersection between the plane <span class="math-container">$x+2y+3z=6$</span> and the surface <span class="math-container">$x^2+y^2=2$</span> and passes through the point <span class="math-container">$(1,1,1)$</span>.</p>
<p>H... | Henry Lee | 541,220 | <p>notice that:
<span class="math-container">$$I(n,k)=\int_0^1\frac{x^k(1-x)^n}{(1-x)^k}dx=\int_0^1 x^k(1-x)^{n-k}dx$$</span>
and we have a definition for the beta function as follows:
<span class="math-container">$$B(a,b)=\int_0^1 x^{a-1}(1-x)^{b-1}dx=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$$</span>
and so:
<span class... |
1,862,807 | <blockquote>
<p>Show that for <span class="math-container">$x,y,z\in\mathbb{Z}$</span>, if <span class="math-container">$x$</span> and <span class="math-container">$y$</span> are coprime and <span class="math-container">$z$</span> is nonzero, then <span class="math-container">$\exists n\in\mathbb{Z}$</span> such that... | André Nicolas | 6,312 | <p>We give an elementary proof that does not use Dirichlet's Theorem. </p>
<p>Let $P$ be the product of the primes that divide $z$ but do not divide $x$. (Recall that an empty product is equal to $1$.) </p>
<p>Since $x$ and $P$ are relatively prime, there is a solution $n$ of the congruence
$$xn\equiv -y+1\pmod{P}.$... |
1,862,807 | <blockquote>
<p>Show that for <span class="math-container">$x,y,z\in\mathbb{Z}$</span>, if <span class="math-container">$x$</span> and <span class="math-container">$y$</span> are coprime and <span class="math-container">$z$</span> is nonzero, then <span class="math-container">$\exists n\in\mathbb{Z}$</span> such that... | darij grinberg | 586 | <p>This is an annoyingly nontrivial statement, despite its harmless sound.
I have two proofs lying around from a number theory homework set that
contained this exercise, so let me post them here; I am sorry for the
mismatching notations.</p>
<blockquote>
<p><strong>Problem 1.</strong>
Let <span class="math-contain... |
1,355,901 | <p>Let $A$ be the set of all integers $x$ such that $x = 2k$ for some integer $k$</p>
<p>Let $B$ be the set of all integers $x$ such that $x = 2k+2$ for some integer $k$</p>
<p>Give a formal proof that $A = B$.</p>
| Jonathan Hebert | 120,932 | <p>You need to show us some effort in the future.</p>
<p>First, to show two sets are equal, we normally pick an element of the first set, show it is contained in the second, then pick an element in the second, and show it is contained in the first.</p>
<p>If we suppose $x \in A$, then $x=2k$ for some integer $k$. Sin... |
2,451,092 | <p>I want to solve a Lagrange multiplier problem,</p>
<p>$$f(x,y) = x^2+y^2+2x+1$$
$$g(x,y)=x^2+y^2-16 $$</p>
<p>Where function $g$ is my constraint.
$$f_x=2x+2, \ \ \ f_y=2y, \ \ \ g_x=2x\lambda, \ \ \ g_y=2y\lambda$$</p>
<p>$$
\begin{cases}
2x+2=2x\lambda \\
2y=2y\lambda \\
x^2+y^2-16=0
\end{cases}
$$</p>
<p>See... | ashi | 299,991 | <p><img src="https://i.stack.imgur.com/0RK0z.jpg" alt="">
this method was taught in our class
Hope this could help you</p>
|
254,623 | <p>For example 100 is even and 100/2= 50 is also even</p>
<p>But 30 is also even but 30/2=15 is odd</p>
<p>Now let's say I have a number as large as 10^10000000000...</p>
<p>I want to know how many steps are involved in cutting this number in half. When the number is even, I divide it by 2. When it's odd, I subtract... | Hagen von Eitzen | 39,174 | <p>Your sequence corresponds quirte directly with the binary representation of the original number: Starting form the least significant binary digit, each $0$ corresponds to "even, dide by two" and each $1$ corresponds to "odd, subtract one; even, divide by two".
For your example $n=100$, which is $1100100$ in binary w... |
860,247 | <p>Simplify $$\frac{3x}{x+2} - \frac{4x}{2-x} - \frac{2x-1}{x^2-4}$$</p>
<ol>
<li><p>First I expanded $x²-4$ into $(x+2)(x-2)$. There are 3 denominators. </p></li>
<li><p>So I multiplied the numerators into: $$\frac{3x(x+2)(2-x)}{(x+2)(x-2)(2-x)} - \frac{4x(x+2)(x-2)}{(x+2)(x-2)(2-x)} - \frac{2x-1(2-x)}{(x+2)(x-2)(2-x... | Omran Kouba | 140,450 | <p>Note that
$$x^2\sqrt{x+x^2}=\frac{x^3+x^4}{\sqrt{x(1+x)}}$$
So, let us look for a polynomial $P(x)=a x^3+bx^2+cx+d$ such that the derivative
$(P(x)\sqrt{x(1+x)})^\prime$ is as close as we can to this function. An easy calculation shows that
$$
\left(P(x)\sqrt{x(1+x)}\right)^\prime=\frac{4 a x^4+(\frac{7 a }{2}+3 b)... |
1,378,536 | <p>Here is a question that naturally arose in the study of some specific integrals. I'm curious if for such integrals are known <em>nice real analysis tools</em> for calculating them (<em>including here all possible sources<br>
in literature that are publicaly available</em>). At some point I'll add my <em>real analy... | M.N.C.E. | 178,187 | <p><strong>Approach 1:</strong></p>
<p>For the first integral
\begin{align}
2\int^1_{0}\frac{{\rm d}x}{\pi^2+\ln^2\left(\frac{1-x}{1+x}\right)}
&=-\frac{4}{\pi}\mathrm{Im}\int^1_0\frac{{\rm d}x}{(\ln{x}+\pi i)(1+x)^2}\tag1\\
&=-\frac{4}{\pi}\mathrm{Im}\int^1_{-1}\frac{{\rm d}x}{\ln{x}(1-x)^2}\tag2\\
&=\fra... |
316,878 | <p>Why is the function not analytic in the complex plane? I believe it is analytic on real plane.</p>
<p>$e^{(-\frac{1}{z^2})}$ where $z\in\mathbb{C}$. </p>
<p>Well a complex function should be infinitely differentiable and should converge. this happens on real plane. But what happens in complex plane?</p>
| Mhenni Benghorbal | 35,472 | <p><strong>Hint:</strong> A function is analytic if and only if its Taylor series about $x_0$ converges to the function in some neighborhood for every $x_0$ in its domain.</p>
|
1,989,291 | <p>What is the closed form of the following:</p>
<p>$$\sum_{j=1}^n 3^{j+1}$$</p>
<p>I'm new to summations. Is it this?</p>
<p>$$\sum_{j=1}^n 3^{j} + \sum_{j=1}^n 3$$</p>
<p>Then using the closed form formula:</p>
<p>$$\frac{3^{n+1} - 1}{2} + 3n$$</p>
| MrYouMath | 262,304 | <p>Hint: This is the finite geometric series.</p>
<p>$$S_0=3\sum_{j=1}^{n}3^j=3S_n$$</p>
<p>We will only look at $S_n$ from now on. $S_n=3+3^2+3^3+\dots+3^n$ so multiply with 3 to get $3S_n=3^2+3^3+\dots+3^{n+1}$. Subtract both equations and notice the chancellations to get $S_n-3S_n=3-3^{n+1}$. Solve for $S_n$ and r... |
1,989,291 | <p>What is the closed form of the following:</p>
<p>$$\sum_{j=1}^n 3^{j+1}$$</p>
<p>I'm new to summations. Is it this?</p>
<p>$$\sum_{j=1}^n 3^{j} + \sum_{j=1}^n 3$$</p>
<p>Then using the closed form formula:</p>
<p>$$\frac{3^{n+1} - 1}{2} + 3n$$</p>
| Bernard | 202,857 | <p>No. Just factor out $3^2$ so as to obtain the standard geometric series:
$$ \sum_{j=1}^n 3^{j+1}=9 \sum_{j=0}^{n-1}3^j=9\frac{3^n-1}{3-1}=\frac{3^{n+2}-3^2}{2}. $$</p>
|
1,436,215 | <p>I'm using the following algorithm (in C) to find if a point lays within a given polygon</p>
<pre><code>typedef struct {
int h,v;
} Point;
int InsidePolygon(Point *polygon,int n,Point p)
{
int i;
double angle=0;
Point p1,p2;
for (i=0;i<n;i++) {
p1.h = polygon[i].h - p.h;
p1.v = polygo... | gammatester | 61,216 | <p>The angle is computed with the <code>atan2</code> function (see
<a href="https://en.wikipedia.org/wiki/Arctan2" rel="nofollow">https://en.wikipedia.org/wiki/Arctan2</a>). But the code is not computationally
robust. The mathematical idea may be OK, but since $\pi$ and $2\pi$ are not exactly
representable as floating ... |
319,058 | <p>Denote <span class="math-container">$\square_m=\{\pmb{x}=(x_1,\dots,x_m)\in\mathbb{R}^m: 0\leq x_i\leq1,\,\,\forall i\}$</span> be an <span class="math-container">$m$</span>-dimensional cube.</p>
<p>It is all too familiar that <span class="math-container">$\int_{\square_1}\frac{dx}{1+x^2}=\frac{\pi}4$</span>.</p>
... | Carlo Beenakker | 11,260 | <p>Yes, this is true, it follows from formulas in <A HREF="https://projecteuclid.org/download/pdf_1/euclid.em/1317758103" rel="noreferrer">Higher-Dimensional Box Integrals</A>, by Jonathan M. Borwein, O-Yeat Chan, and R. E. Crandall. </p>
<p><span class="math-container">\begin{align}
&\text{define}\;\;C_{m}(s)=\in... |
2,551,683 | <blockquote>
<p>A jet has a $5\%$ chance of crashing on any given test flight. Once it crashes the program will be halted. Find the probability that the program lasts less than three flights.</p>
</blockquote>
<p>The correct answer to this question is $0.1426$, but I can't figure out how to get it.</p>
<p>Here's my... | Graham Kemp | 135,106 | <p>$pq^{x-1}$ is the probability for $x$ <em>trials until</em> the first success, where $x\in\{1,2,\ldots\}$.</p>
<p>$pq^x$ is the probability for $x$ <em>failures before</em> the first success, where $x\in\{0,1,2,\ldots\}$.</p>
<p>Unfortunately both distributions are called Geometric but it important to not confuse ... |
1,791,673 | <p>I was wondering about this, just now, because I was trying to write something like:<br>
$880$ is not greater than $950$. <br>
I am wondering this because there is a 'not equal to': $\not=$ <br>
Not equal to is an accepted mathematical symbol - so would this be acceptable: $\not>$? <br>
I was searching around but ... | DomS | 518,038 | <p>I would just like to make it clear that ≮ is NOT the same as ≥</p>
<p>Here is an example: </p>
<p>1+²≮(1+)² is clearly not the same as 1+²≥(1+)²</p>
<p>Think of =-0.5 and =2 as examples to highlight this, because although when =-0.5, 1+²≥(1+)² but when =2, 1+²≱(1+)²</p>
<p>Therefore, think of ≮ as meaning "not g... |
177,515 | <p>From <a href="http://mitpress.mit.edu/algorithms/" rel="nofollow">Cormen et all</a>:</p>
<blockquote>
<p>The elements of a matrix or vectors are numbers from a number system, such as the real numbers , the complex numbers , or integers modulo a prime .</p>
</blockquote>
<p>What do they mean by <strong>integers m... | Andrea Mori | 688 | <p>You can actually form matrices with entries in any <a href="http://en.wikipedia.org/wiki/Ring_%28mathematics%29" rel="nofollow">ring</a>, although sometimes you won't have the same nice properties.</p>
<p>The ring of integers modulo a prime, sometimes denoted $\Bbb F_p$, is the ring where you perform <a href="http:... |
951,332 | <p>the integer 220, 251 304 represent three consecutive perfect squares in base b. Determine the value of b.</p>
| Mark Bennet | 2,906 | <p>Hint: the differences between three consecutive squares are two consecutive odd integers.</p>
<blockquote class="spoiler">
<p> The first difference is $31$ so the second will be $33$ which means that $25+3=30$ and $b=8$</p>
</blockquote>
|
951,332 | <p>the integer 220, 251 304 represent three consecutive perfect squares in base b. Determine the value of b.</p>
| Jack D'Aurizio | 44,121 | <p>We have that $3b+1$ and $b^2-5b+3$ are two consecutive odd integers, hence $b^2-8b+2=2$, so the only possibility is $b=8$, and it fits.</p>
|
3,290,839 | <p>I want to prove that that given <span class="math-container">$f:R^2 \rightarrow R$</span> which is continuous with compact support s.t the integral of <span class="math-container">$f$</span> for every straight line <span class="math-container">$l$</span> is zero (<span class="math-container">$\int f(l(t))\mathrm{d}t... | Redundant Aunt | 109,899 | <p>Suppose there exists <span class="math-container">$p\in\mathbb{R}^2$</span> with <span class="math-container">$f(p)\neq 0$</span>, then WLOG (by replacing <span class="math-container">$f$</span> by <span class="math-container">$-f$</span> if necessary) we might assume <span class="math-container">$f(p)>0$</span>.... |
2,561,125 | <p>Hey having trouble finishing this question.</p>
<p>Prove by induction that $n^3 \le 2^n$ for all natural numbers $n\ge 10$.</p>
<p>This is what I have so far:</p>
<p>Base step: For $n = 10$ </p>
<p>$1000 \le 1024$</p>
<p>Assumption Step: For $n = k$</p>
<p>Assume $k^3 \le 2^k$</p>
<p>Induction step: For $n =... | Robin Grajeda | 512,209 | <p>Define $$S=\left\{ n\in
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
,\text{ }n\geq 10\text{ }|\text{ }n^{3}\leq 2^{n}\right\} $$</p>
<p>For $n=10$</p>
<p>$$
10^{3}\leq 2^{10}
$$</p>
<p>Therefore $S\neq \varnothing .$</p>
<p>Suppose that $k^{3}\leq 2^{k}$. We want to show that for any positive... |
4,263,784 | <p>I'm trying to find the multiplicity of <span class="math-container">$z=0$</span> on <span class="math-container">$f(z)=z\cos(z)-\sin(z)$</span> using complex analysis.</p>
<p>I'm new to complex analysis and the argument principle/Rouché's theorem so I'm not quite sure where to start. I can prove how many zero's this... | Henno Brandsma | 4,280 | <p>Given a cipher text of length <span class="math-container">$L$</span>, and plain texts of the same length <span class="math-container">$L$</span> are equally likely (as all keys are of the same length and random). Any attacker has no way of telling which is right.</p>
|
194,421 | <p>This is homework. The problem was also stated this way: </p>
<p>Let A be a dense subset of $\mathbb{R}$ and let x$\in\mathbb{R}$. Prove that there exists a decreasing sequence $(a_k)$ in A that converges to x.</p>
<p>I know:</p>
<p>A dense in $\mathbb{R}$ $\Rightarrow$ every point in $\mathbb{R}$ is either in A o... | BaronVT | 39,526 | <p>There is a sequence converging to $x$, but you won't know it's decreasing - you'll have to construct it so this happens.</p>
<p>Try thinking about what would happen if this weren't true - then there would be an $a_0 \in A$ so that $(x,a_0) \cap A = \emptyset$... in other words this interval is full of points that a... |
459,579 | <blockquote>
<p>Find the value of $3^9\cdot 3^3\cdot 3\cdot 3^{1/3}\cdot\cdots$</p>
</blockquote>
<p>Doesn't this thing approaches 0 at the end? why does it approaches 1?</p>
| lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>Using <a href="http://www.proofwiki.org/wiki/Exponent_Combination_Laws" rel="nofollow">Exponent Combination Laws</a>,
$$a^m\cdot a^n\cdot a^p\cdots=a^{m+n+p+\cdot},$$</p>
<p>$$\displaystyle 3^9\cdot 3^3\cdot3\cdot 3^\frac13\cdots=3^{\left(3^2+3+1+\frac13+\cdots\right)}$$</p>
<p>Observe that the power... |
2,961,023 | <p>Is it allowed to solve this inequality <span class="math-container">$x|x-1|>-3$</span> by dividing each member with <span class="math-container">$x$</span>? What if <span class="math-container">$x$</span> is negative?</p>
<p>My textbook provides the following solution:</p>
<blockquote>
<p>Divide both sides b... | nonuser | 463,553 | <p>For <span class="math-container">$x\geq0$</span> this inequality is always true. </p>
<p>Assume that <span class="math-container">$x<0$</span>, so <span class="math-container">$x=-y$</span> for some positive <span class="math-container">$y$</span> and we get <span class="math-container">$$y|\;\underbrace{y+1}_{&... |
2,475,507 | <blockquote>
<p>Find $f$ and $g$ such that domain $(f\circ g)=\mathbb{R}$ and domain $(g\circ f)=\emptyset$</p>
</blockquote>
<p>That's it, I can't think of any. </p>
<p>I've thought of $f(x)=-1$ and $g(x)=\sqrt{x}$, and then: $$f\big(g(x)\big)=-1$$ $$g\big(f(x)\big)=\sqrt{-1}$$ </p>
<p>Which would in principle sa... | Andreas Blass | 48,510 | <p>You already have a correct answer, from user334639, under the assumption that all the inputs and outputs of your functions are real numbers. On the other hand, if you're allowed to use some entity $Q$ that isn't a real number, then you can obtain an example. Let $g$ be the identity function on $\mathbb R$ and let $... |
365,287 | <p>Let $([0,1],\mathcal{B},m)$ be the Borel sigma algebra with lebesgue measure and $([0,1],\mathcal{P},\mu)$ be the power set with counting measure. Consider the product $\sigma$-algebra on $[0,1]^2$ and product measure $m \times \mu$.</p>
<p>(1) Is $D=\{(x,x)\in[0,1]^2\}$ measurable?</p>
<p>(2) If so, what is $m \t... | Brian M. Scott | 12,042 | <p>Charles Wells, <em>The Handbook of Mathematical Discourse</em>; it’s available as a PDF <a href="https://www.abstractmath.org/Handbook/handbook.pdf" rel="nofollow noreferrer">here</a>. His site <a href="http://www.abstractmath.org/MM/MMIntro.htm" rel="nofollow noreferrer">abstractmath.org</a> may also be useful.</p>... |
2,541,991 | <p>I need to find a pair of dependent random variables $(X, Y)$ with covariance equal to $0.$ From this I gather:</p>
<p>$$0 = E((X-EX)(Y-EY)) = E \left(\left(X - \int_{-\infty}^\infty xf_X(x)\,dx\right) \left(Y - \int_{-\infty}^\infty xf_Y(x)\,dx \right)\right)$$</p>
<p>but what can I do now? How can I use the fact ... | Remy | 325,426 | <p>Let $X$ be a random variable that takes on the values $1$ or $-1$ with equal probability. Let $Y$ be a random variable where $Y=0$ if $X=-1$, and $Y$ is $-1$ or $1$ with equal probability if $X=1$.</p>
<p>Then $X$ and $Y$ depend on each other, since if you know what $Y$ is then you know what $X$ is. However, thei... |
2,331,961 | <p>Please suggest: </p>
<h1>Question 1)</h1>
<p>What are some reasonable assumptions regarding the limit of the Cumulative Distribution as the Variance grows to infinity.</p>
<p>$$
\lim_{\sigma\rightarrow\infty} F\left(t,\sigma\right) = \text{??}
$$</p>
<h1>Question 2)</h1>
<p>Also, is it a reasonable assumption t... | Michael Hardy | 11,667 | <p>One example is the normal distribution with expected value $0.$</p>
<p>Let $Z\sim N(0,1).$ Then $\sigma Z\sim N(0,\sigma^2).$ Let $F$ be the c.d.f. of $\sigma Z.$ Then
$$
F(t) = \Pr(\sigma Z\le t) = \Pr\left( Z \le \frac t \sigma \right).
$$
As $\sigma$ increases, then $t/\sigma\quad \begin{cases} \text{increases t... |
2,331,961 | <p>Please suggest: </p>
<h1>Question 1)</h1>
<p>What are some reasonable assumptions regarding the limit of the Cumulative Distribution as the Variance grows to infinity.</p>
<p>$$
\lim_{\sigma\rightarrow\infty} F\left(t,\sigma\right) = \text{??}
$$</p>
<h1>Question 2)</h1>
<p>Also, is it a reasonable assumption t... | texmex | 238,328 | <p>Adding a few more examples for Question 2, in addition to the clarifications by Robert Israel and Michael Hardy.</p>
<h1>Example 1:</h1>
<p>The derivative of the cumulative distribution of a uniformly distributed
variable, having support $t\in\left[a,b\right]\;;0\leq a,b<\infty$,
with respect to the standard de... |
1,151,726 | <p>The following question is from Fred H. Croom's book "Principles of Topology"</p>
<blockquote>
<blockquote>
<p>In <span class="math-container">$\mathbb{R}^n$</span>, let <span class="math-container">$R$</span> denote the set of points having only rational coordinates and <span class="math-container">$I$</sp... | user4894 | 118,194 | <p>Yes, your part 1 proof is good. </p>
<p>But now you're already 90% of the way to #2. If every ball contains points of both $R$ and $I$, what can you say about the set of limits points of $R$ and $I$, respectively?</p>
<p>And 3 is just another minor variation on this same theme. What's the definition of the boundar... |
334,075 | <p>How do you compute $$\int_{0}^1 \frac{\arctan x }{1+x} dx$$</p>
| Santosh Linkha | 2,199 | <p>Using integration by parts
$$\int_0^1 \frac{\arctan x}{1+x} dx = \arctan(x) \ln(1+x)|_0^1 - \int_0^1 \frac{\ln (1+x)}{1+x^2}dx$$</p>
<p>The former part is $\displaystyle \frac{\pi}{
4} \ln 2 $ and the latter part is $\displaystyle \frac{\pi}{8} \ln 2$ which is answered <a href="https://math.stackexchange.com/que... |
334,075 | <p>How do you compute $$\int_{0}^1 \frac{\arctan x }{1+x} dx$$</p>
| robjohn | 13,854 | <p>$$
\begin{align}
\int_0^1\frac{\arctan(x)}{1+x}\,\mathrm{d}x
&=\int_0^{\pi/4}\frac{\theta}{1+\tan(\theta)}\,\sec^2(\theta)\,\mathrm{d}\theta\tag{1}\\[6pt]
&=\int_0^{\pi/4}\frac{\theta\,\mathrm{d}\theta}{\cos(\theta)\,(\cos(\theta)+\sin(\theta))}\tag{2}\\[6pt]
&=\int_0^{\pi/4}\frac{(\frac\pi4-\theta)\,\ma... |
164,002 | <p>When I am reading a mathematical textbook, I tend to skip most of the exercises.
Generally I don't like exercises, particularly artificial ones.
Instead, I concentrate on understanding proofs of theorems, propositions, lemmas, etc..</p>
<p>Sometimes I try to prove a theorem before reading the proof.
Sometimes I try... | benny rimmer | 35,048 | <p>I'm with the OP on this, I skip exercises too. </p>
<p>Here's the logic: A true understanding of maths is about being creative in its applications and not just the material initself. </p>
<p>Exercises, by definition, stifle creativity by presenting a sandbox in which to think. </p>
<p>It's a bit like Rocky 3, do ... |
2,313,060 | <p>$f(\bigcap_{\alpha \in A} U_{\alpha}) \subseteq \bigcap_{\alpha \in A}f(U_{\alpha})$</p>
<p>Suppose $y \in f(\bigcap_{\alpha \in A} U_{\alpha})$
$\implies f^{-1}(y) \in \bigcap_{\alpha \in A} U_{\alpha} \implies f^{-1}(y) \in U_{\alpha}$ for all $\alpha \in A$</p>
<p>$\implies y \in f (U_{\alpha})$ for all $\alph... | Angina Seng | 436,618 | <p>There is no "$f^{-1}(y)$" in general. Although by definition $y$
is equal to $f(x)$ for some $x\in\bigcap U_\alpha$, that $x$ might
not be unique. So take an $x\in\bigcap U_\alpha$ and repeat your
argument with $x$ replacing "$f^{-1}(y)$".</p>
<p>You then attempt to prove the reverse inclusion: $\bigcap f(U_\alpha)... |
2,313,060 | <p>$f(\bigcap_{\alpha \in A} U_{\alpha}) \subseteq \bigcap_{\alpha \in A}f(U_{\alpha})$</p>
<p>Suppose $y \in f(\bigcap_{\alpha \in A} U_{\alpha})$
$\implies f^{-1}(y) \in \bigcap_{\alpha \in A} U_{\alpha} \implies f^{-1}(y) \in U_{\alpha}$ for all $\alpha \in A$</p>
<p>$\implies y \in f (U_{\alpha})$ for all $\alph... | egreg | 62,967 | <p>You are probably misled by the similar relation holding for $f^{-1}$:
$$
\bigcap_\alpha f^{-1}(U_\alpha)=
f^{-1}\Bigl(\bigcap_\alpha U_\alpha\Bigr)
$$
which <em>is</em> true and whose proof goes essentially like yours, with $f$ and $f^{-1}$ interchanged.</p>
<p>However, your assignment asks you to find an example o... |
939,725 | <p>Given that $a_0=2$ and $a_n = \frac{6}{a_{n-1}-1}$, find a closed form for $a_n$.</p>
<p>I tried listing out the first few values of $a_n: 2, 6, 6/5, 30, 6/29$, but no pattern came out. </p>
| Pauly B | 166,413 | <p>Start with $a_n=\frac6{a_{n-1}-1}$, and replace $a_{n-1}$ with $\frac6{a_{n-2}-1}$. We obtain</p>
<p>$$a_n=\frac6{\frac6{a_{n-2}-1}-1}=\frac{6(1-a_{n-2})}{a_{n-2}-7}$$</p>
<p>Doing this again with $a_{n-2}=\frac6{a_{n-3}-1}$ and so forth, we get</p>
<p>$$a_n=\frac{6(7-a_{n-3})}{7a_{n-3}-13}=\frac{6(13-7a_{n-4})}{... |
1,075,879 | <p>I have to prove or disprove the following statement:</p>
<blockquote>
<p>If a group $G$ acts on a set $X$, then every subgroup $H$ of $G$ acts on the set $X$ as well, and every orbit of the action $G$ on $X$ is an union of orbits of the action $H$ on $X$.'</p>
</blockquote>
<p>But I have absolutely no clue what ... | user133281 | 133,281 | <p>In general, an action of a group $G$ on a set $X$ is a group homomorphism from $G$ to the group $S_X$ of permutations of the set $X$. This means that we send each group element $g \in G$ to some permutation of the elements in $X$, so each group element "acts" on $X$ by permuting its elements in some way. Usually, we... |
1,075,879 | <p>I have to prove or disprove the following statement:</p>
<blockquote>
<p>If a group $G$ acts on a set $X$, then every subgroup $H$ of $G$ acts on the set $X$ as well, and every orbit of the action $G$ on $X$ is an union of orbits of the action $H$ on $X$.'</p>
</blockquote>
<p>But I have absolutely no clue what ... | Community | -1 | <p>As for the second part, recall that every <span class="math-container">$g\in G$</span> lays in some right coset of <span class="math-container">$H$</span> in <span class="math-container">$G$</span>. So, denoted with <span class="math-container">$R\subseteq G$</span> a complete set of coset representatives, for <span... |
3,177,343 | <p>I have the following minimization problem in <span class="math-container">$x \in \mathbb{R}^n$</span></p>
<p><span class="math-container">$$\begin{array}{ll} \text{minimize} & \|x\|_2 - c^T x\\ \text{subject to} & Ax = b\end{array}$$</span></p>
<p>where <span class="math-container">$A \in \mathbb{R}^{m \t... | Community | -1 | <p>Let <span class="math-container">$A^+$</span> be the Moore Penrose inverse of <span class="math-container">$A$</span>. Note that <span class="math-container">$x=x_0+u$</span> where <span class="math-container">$u$</span> varies in the image of the symmetric matrix <span class="math-container">$I_n-A^+A$</span>, a sp... |
3,177,343 | <p>I have the following minimization problem in <span class="math-container">$x \in \mathbb{R}^n$</span></p>
<p><span class="math-container">$$\begin{array}{ll} \text{minimize} & \|x\|_2 - c^T x\\ \text{subject to} & Ax = b\end{array}$$</span></p>
<p>where <span class="math-container">$A \in \mathbb{R}^{m \t... | greg | 357,854 | <p>The linear equation <span class="math-container">$Ax=b$</span> has the general solution
<span class="math-container">$$x = A^+b + Pw$$</span>
where <span class="math-container">$A^+$</span> is the Penrose inverse, <span class="math-container">$P=(I-A^+A)$</span> is the projector into the nullspace, and <span class="... |
82,765 | <p><strong>Bug introduced in 9.0 and persisting through 12.2</strong></p>
<hr />
<p>I get the following output with a fresh Mathematica (ver 10.0.2.0 on Mac) session</p>
<pre><code>FullSimplify[Exp[-100*(i-0.5)^2]]
(* 0. *)
Simplify[Exp[-100*(i-0.5)^2]]
(* E^(-100. (-0.5+i)^2) *)
</code></pre>
<p><code>FullSimplif... | David Zwicker | 26,656 | <p>The behavior seems to be a bug of Mathematica.
Here is an excerpt from an email I got from Wolfram after asking them about the problem:</p>
<blockquote>
<p>It does seem that the answer of FullSimplify is incorrect especially
since the exponential function is not identical to zero (or a
very-close-to-zero cons... |
4,441,034 | <blockquote>
<p>Consider function <span class="math-container">$f(x)$</span> whose derivative is continuous on the interval <span class="math-container">$[-3; 3]$</span> and the graph of the function <span class="math-container">$y = f'(x)$</span> is pictured below. Given that <span class="math-container">$g(x) = 2f(x)... | heropup | 118,193 | <p>Let's look at an example. Say <span class="math-container">$n = 12$</span>. Then for <span class="math-container">$k \in \{1, \ldots, 12\}$</span>, we look at set of divisors of <span class="math-container">$k$</span>:</p>
<p><span class="math-container">$$\begin{array}{c|l|c}
k & d & d(k)\\
\hline
1 &... |
4,441,034 | <blockquote>
<p>Consider function <span class="math-container">$f(x)$</span> whose derivative is continuous on the interval <span class="math-container">$[-3; 3]$</span> and the graph of the function <span class="math-container">$y = f'(x)$</span> is pictured below. Given that <span class="math-container">$g(x) = 2f(x)... | PNT | 873,280 | <p>Let's make a table, the rows and columns are <span class="math-container">$1,2,3...,n$</span> and an element <span class="math-container">$(d,k)$</span> is <span class="math-container">$1$</span> if <span class="math-container">$d\mid k$</span> and <span class="math-container">$0$</span> otherwise,
<span class="math... |
3,830,231 | <p>I'm trying to prove the following 'covariance inequality'
<span class="math-container">$$
|\text{Cov}(x,y)|\le\sqrt{\text{Var}(x)}\sqrt{\text{Var}(y)}\,,
$$</span>
where covariance and variance are defined using discrete values,
<span class="math-container">$$
\text{Cov}(x,y) = \frac{1}{n-1}\sum_{i=1}^n \big[(x_i-\b... | User203940 | 333,294 | <p>First recall that</p>
<p><span class="math-container">$$ \text{Var}(x) = \frac{1}{n} \sum_{j=0}^{n-1} (x_i - \bar{x})^2.$$</span></p>
<p>See for example <a href="https://en.wikipedia.org/wiki/Variance#Discrete_random_variable" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Variance#Discrete_random_variable<... |
2,830,718 | <p>I want to estimate how many red balls in a box. Red, yellow, blue balls could be in the box. But I don't know how many of them are in the box.</p>
<p>What I did was randomly drawing 10 balls from the box and learned that there was no red ball.</p>
<p>(Edit: Assume the number of the balls in the box is a known fini... | David K | 139,123 | <p>TL;DR: The answer to the title of your question is, "Yes, but not with anywhere near the kind of accuracy and confidence you seem to be asking for."</p>
<hr>
<p>Let's say I put $10000$ balls in a box, and exactly one of the balls is red.
You know the total number of balls but not how many there are of each color.<... |
4,188,106 | <p>Let's say we have the following diagram
<span class="math-container">$$\require{AMScd}\begin{CD}
0 @>>> A @>>> B @>>> C @>>> 0\\
{} @V{\alpha}VV @V{\beta}VV @V{\gamma}VV {} \\
0 @>>> A' @>>> B' @>>> C' @>>> 0
\end{CD}$$</span>
where the top and... | Milten | 620,957 | <p>Let <span class="math-container">$(a,b)=\beta^{-1}(1,0)$</span>. Then <span class="math-container">$(a,b)$</span> has order <span class="math-container">$4$</span> in <span class="math-container">$B$</span>, so <span class="math-container">$a$</span> is odd. Therefore
<span class="math-container">$$
g(a,b) = (1,b) \... |
2,550,568 | <p>Suppose we have an alphabet of $a$ letters and a word $w$ of length $r$. What is the probablity that $w$ will appear in a sequence of $n$ letters drawn at random from the given alphabet?</p>
<p>I have posted a general question since there seem to be a few of these questions appearing, and this is intended as a gene... | Mark Bennet | 2,906 | <p>Let $a_k$ be the probability that the word appears in the first $k$ letters. We have $a_0=a_1= \dots =a_{r-1}=0$.</p>
<p>Either $w$ appears in the first $n-1$ letters chosen, or it appears for the first time at the $n^{th}$ digit. In this second case (excluding the case of overlaps - see below) the final $r$ digits... |
4,765 | <p>I have a grid made up of overlapping <span class="math-container">$3\times 3$</span> squares like so:</p>
<p><img src="https://i.stack.imgur.com/BaY9s.png" alt="Grid"></p>
<p>The numbers on the grid indicate the number of overlapping squares. Given that we know the maximum number of overlapping squares (<span clas... | Alan D'Souza | 74,758 | <p>Try proving that for any irrational number $\alpha$, the set
$A=\left \{ a+b\alpha \mid a\in \mathbb{N},b\in \mathbb{Z} \right \}$ is dense in $\mathbb{R}$. Let $\alpha =\pi $.
Since the set $A$ is dense in $\mathbb{R}$, $\forall \; x\in \mathbb{R}$, there exists a sequence of terms $(z_{n})$ such that
$\lim_{n\ri... |
1,198,373 | <p>I have a function of two variables, which I wish to check for monotonicity in the entire function domain. I cant find any formal definition of increasing or decreasing function for multi variable case. Can anybody please guide?
Thanks in advance.</p>
| Clement C. | 75,808 | <p>There is no general definiton (as mentioned in the comments, there is no total order on <span class="math-container">$\mathbb{R}^2$</span>, which would be required for a canonical definition of monotonicity of bivariate functions).</p>
<p>Two <em>possible</em> definitions, though: let <span class="math-container">$$... |
1,198,373 | <p>I have a function of two variables, which I wish to check for monotonicity in the entire function domain. I cant find any formal definition of increasing or decreasing function for multi variable case. Can anybody please guide?
Thanks in advance.</p>
| Ij Huij | 495,418 | <p>I wanted to add this as a comment for the first answer, but I cannot put comments. The first and the second definition in this answer are equivalent.</p>
<p>If $f$ satisfies the first definition then it satisfies the second. We can see that by sandwiching $f(x',y)$ (or $f(x,y')$) between $f(x,y)$ and $f(x',y')$.</p... |
1,390,093 | <p>Let $G$ be act on $\Gamma$ with a fundamental domain $T$ where $T$ is tree. We construct <em>tree of groups</em> $(\mathcal{G},T)$ with the following structure:
$$\text{for every } v\in V(T),\,\,G_v=\operatorname{Stab}_G(v) $$</p>
<p>$$\text{for every } e\in E(T),\,\,G_e=\operatorname{Stab}_G(e) $$</p>
<p>Assume t... | Lee Mosher | 26,501 | <p>We cannot conclude that $\phi$ is surjective, because it is not true in general. Here is a counterexample. </p>
<ul>
<li>$\Gamma = \mathbb{R}$ with vertex set $\mathbb{Z}$. </li>
<li>$G = \mathbb{Z}$ acting on $\mathbb{R}$ by translation. </li>
<li>$T = [0,1]$ is a fundamental domain. </li>
</ul>
<p>The stabilizer... |
3,840,253 | <blockquote>
<p>How to show that <span class="math-container">$\csc x - \csc\left(\frac{\pi}{3} + x \right) + \csc\left(\frac{\pi}{3} - x\right) = 3 \csc 3x$</span>?</p>
</blockquote>
<p>My attempt:<br />
<span class="math-container">\begin{align}
LHS &= \csc x - \csc\left(\frac{\pi}{3} + x\right) + \csc\left(\frac... | Michael Rozenberg | 190,319 | <p><span class="math-container">$$\frac{1}{\sin{x}}-\frac{1}{\sin\left(\frac{\pi}{3}+x\right)}+\frac{1}{\sin\left(\frac{\pi}{3}-x\right)}=$$</span>
<span class="math-container">$$=\frac{\sin\left(\frac{\pi}{3}-x\right)\sin\left(\frac{\pi}{3}+x\right)+\sin{x}\left(\sin\left(\frac{\pi}{3}+x\right)-\sin\left(\frac{\pi}{3}... |
4,082,588 | <blockquote>
<p><strong>Definition:</strong> <span class="math-container">$\beta X$</span> is the Stone-Čech compactification of <span class="math-container">$X$</span>.</p>
</blockquote>
<blockquote>
<p><strong>Theorem A:</strong> If <span class="math-container">$K$</span> is a compact Hausdorff space and <span class=... | Peluso | 884,108 | <p>This proof looks correct, you just need a little push.</p>
<p>We will use the following result:</p>
<p><strong>Lemma:</strong> If <span class="math-container">$X$</span> is a topological space, <span class="math-container">$D$</span> is a dense subset of <span class="math-container">$X$</span> and <span class="math-... |
2,236,717 | <p>Let $S$ be a regular domain of characteristic $p>0$ with fraction field $K$. Assume that $K$ is $F$-finite, meaning that $K$ is a finite module over $K^p$. Does it follow that $S$ is also $F$-finite?</p>
<p>Diego</p>
| Takumi Murayama | 116,766 | <p>I believe this is false. [<a href="http://dx.doi.org/10.2140/ant.2016.10.1057" rel="nofollow noreferrer">Datta–Smith</a>, Ex. 4.5.1] give an example of a DVR that is not $F$-finite, whose fraction field is $\mathbf{F}_p(x,y)$.</p>
<p>A way to produce more examples is the following:</p>
<p><strong>Proposition</stro... |
4,342,737 | <p>My question: If you throw a dice 5 times, what is the expected value of the square of the median of the 5 results?</p>
<p>A slightly modified question would be: If you throw a dice 5 times, what is the expected value of the median? The answer would be 3.5 by symmetry.</p>
<p>For the square, it seems to be that symme... | Masacroso | 173,262 | <p>Let <span class="math-container">$X_1,\ldots, X_n$</span> be i.i.d. r.v. If we order the previous list of r.v. by it values we get r.v. <span class="math-container">$X_{(1)},\ldots ,X_{(n)}$</span> named the ranks of the list <span class="math-container">$X_1,\ldots,X_n$</span>. Now, observe that the distribution of... |
2,949,789 | <p>Suppose I have some function <span class="math-container">$V(x)=x+log(c)$</span>, where <span class="math-container">$x$</span> is a continuous random variable and <span class="math-container">$c$</span> a constant bounded on <span class="math-container">$[0,1]$</span>. I have some queries regarding the following:</... | Community | -1 | <p><span class="math-container">$$\frac{dV(x)}{dx}=1$$</span></p>
<p>Randomness of <span class="math-container">$x$</span> does not matter. (And in this particular case, even the value of <span class="math-container">$x$</span> does not matter.)</p>
|
67,513 | <p>When processing a larger Dataset I came up do a point where I want to form a dataset with culumn heads from an intermediate structure. Here is an example of this structure:</p>
<pre><code>test = {<|"name" -> "alpha",
"group" -> "one"|> -> {<|"value" -> 459|>}, <|"name" -> "beta",... | Karsten 7. | 18,476 | <pre><code>test = {<|"name" -> "alpha", "group" -> "one"|> -> {<|"value" -> 459|>},
<|"name" -> "beta", "group" -> "two"|> -> {<|"value" -> -338|>},
<|"name" -> "gamma", "group" -> "two"|> -> {<|"value" -> 363|>}};
Association /@... |
67,513 | <p>When processing a larger Dataset I came up do a point where I want to form a dataset with culumn heads from an intermediate structure. Here is an example of this structure:</p>
<pre><code>test = {<|"name" -> "alpha",
"group" -> "one"|> -> {<|"value" -> 459|>}, <|"name" -> "beta",... | kglr | 125 | <pre><code>Dataset[Join@@@({#,Join@@#2}&@@@test)]
(* or Dataset[Join@@@({#,Sequence@@#2}&@@@test)] *)
</code></pre>
<p><img src="https://i.stack.imgur.com/NMa7J.png" alt="enter image description here"></p>
|
74,188 | <blockquote>
<p>Let <span class="math-container">$a,c \in \mathbb R$</span> with <span class="math-container">$a \neq 0$</span>, and let <span class="math-container">$b \in \mathbb C$</span>. Define <span class="math-container">$$S=\{z\in \mathbb C: az\bar{z}+b\bar{z}+\bar{b}z+c=0\}.$$</span></p>
<p>a. Show that <span ... | Gerry Myerson | 8,269 | <p>One way is to work your way backward to the problem from what you know about circles. A circle has a center and a radius. Let's call the center $w$, the radius, $r$. The circle is all the points $z$ whose distance from $w$ is $r$. The square of the distance between two points $u$ and $v$ in the complex plane is $|u-... |
4,046,532 | <p><strong>QUESTION 1:</strong> Let <span class="math-container">$f, g: S\rightarrow \mathbb{R}^m$</span> be differentiable vector-valued functions and let <span class="math-container">$\lambda\in \mathbb{R}$</span>. Prove that the function <span class="math-container">$(f+g):S\rightarrow \mathbb{R}^m$</span> is also d... | Rafael | 894,475 | <p>Question 1. The definition of sum of functions <span class="math-container">$f$</span> and <span class="math-container">$g$</span> is a function <span class="math-container">$(f+g):S\rightarrow \mathbb{R}^m$</span>, which can be described with a formula <span class="math-container">$(f+g)(s)=f(s)+g(s)$</span>. It is... |
2,294,997 | <p>How to prove that $\displaystyle 0,02<\int_0^1 \frac{x^7}{(e^x+e^{-x})\sqrt{1+x^2}}dx<0,05$? I tried to use mean value theorems, but i failed.</p>
| Mark Viola | 218,419 | <p>HINT:</p>
<p>Note that $2\le e^x+e^{-x}\le e+e^{-1}$ and $x\le \sqrt{1+x^2}\le \sqrt{2}$.</p>
|
964,372 | <p>I have a general question.</p>
<p>If there is a matrix which is inverse and I multiply it by other matrixs which are inverse.
Will the result already be reverse matrix?</p>
<p>My intonation says is correct, but I'm not sure how to prove it.</p>
<p>Any ideas? Thanks.</p>
| Petite Etincelle | 100,564 | <p>When you have $n$ people, consider what you do with the last person.</p>
<p>If you let him alone, you have $A_{n-1}$ ways to group the other $n-1$ people.</p>
<p>If you group him with someone else, you have $n-1$ ways to do that. And after that you need to group the other $n-2$ people. That's where comes from $(n-... |
1,828,042 | <p>This is my first question on this site, and this question may sound disturbing. My apologies, but I truly need some advice on this.</p>
<p>I am a sophomore math major at a fairly good math department (top 20 in the U.S.), and after taking some upper-level math courses (second courses in abstract algebra and real an... | treble | 24,837 | <p>Hopefully I can answer before the tide of "talk to someone who knows you personally" and "this question is off topic" rolls in and your question is inevitably closed. It is true that you should talk to someone who knows you better, but I can give you some general advice that is better than just "follow your heart."<... |
2,165,296 | <p>Can every separable Banach space be isometrically embedded in $l^2$ ? Or at least in $l^p$ for some $1\le p<\infty$ ? </p>
<p>I only know that any separable Banach space is isometrically isomorphic to a linear subspace of $l^{\infty}$.</p>
<p>Please help . Thanks in advance </p>
| user90369 | 332,823 | <p>$\displaystyle \frac{1}{1-x}=\sum\limits_{n=0}^\infty x^n\enspace$ for $\enspace-1<x<1$ , proof by multiplication with $1-x$ . </p>
<p>One derivation for $x$ gives $\enspace\displaystyle \frac{1}{(1-x)^2}=\sum\limits_{n=1}^\infty nx^{n-1}$ . </p>
<p>With $x:=-2t$ and therefore $\enspace -\frac{1}{2}< t<... |
1,212,000 | <p>I was trying to solve this square root problem, but I seem not to understand some basics. </p>
<p>Here is the problem.</p>
<p>$$\Bigg(\sqrt{\bigg(\sqrt{2} - \frac{3}{2}\bigg)^2} - \sqrt[3]{\bigg(1 - \sqrt{2}\bigg)^3}\Bigg)^2$$</p>
<p>The solution is as follows:</p>
<p>$$\Bigg(\sqrt{\bigg(\sqrt{2} - \frac{3}{2}\b... | Community | -1 | <p>This is a mistake that might have been avoided by being aware of the numerical values of some of the expressions involved. A very rough approximation for $\sqrt{2}$ is $1.41$. Clearly $\frac{3}{2} = 1.5$. Then $\sqrt{2} - \frac{3}{2} \approx -0.09$, and that squares to $0.0081$, and the square root of that is $0.09$... |
2,661,210 | <p>Let $a_{1}, \dots, a_{n}$ be real numbers not all zero; let $b_{1},\dots, b_{n}$ be real numbers; let $\sum_{1}^{n}b_{i} \neq 0$. Then does there exist real numbers $w_{1},\dots, w_{n} > 0$ such that
$$
\frac{\sum_{1}^{n}w_{i}a_{i}}{\sum_{1}^{n}w_{i}b_{i}} > \frac{\sum_{1}^{n}a_{i}}{\sum_{1}^{n}b_{i}}?
$$
Som... | mucciolo | 222,084 | <p>$\DeclareMathOperator{\spn}{span}\DeclareMathOperator{\img}{Im}$There is really some linear algebra hidden there. Note that $\sum_{i=1}^{n}w_{i}a_{i}$ is a linear functional on $\mathbb{R}^n$ with respect to $w$. That is, $\alpha: \mathbb{R}^n \to \mathbb{R} : (w_1, \cdots, w_n) \mapsto \sum_{i=1}^{n}w_{i}a_{i}$ is ... |
4,629,922 | <p>I'm reading about the method of two-timing in section 7.6 of <em>Nonlinear Dynamics and Chaos</em> by Strogatz, and I have some questions about how to make this concept rigorous. In this section the book considers equations of the form
<span class="math-container">$$ \hspace{4.5cm} \ddot{x} + x + \epsilon h(x,\dot{x... | eyeballfrog | 395,748 | <p>So firstly, they really should be writing <span class="math-container">$X(t, \epsilon t; \epsilon)$</span>. Without that the expansion in terms of <span class="math-container">$x_i(\tau, T)$</span> doesn't quite make sense. But as you note, it <em>does</em> make sense that for any function <span class="math-containe... |
4,629,922 | <p>I'm reading about the method of two-timing in section 7.6 of <em>Nonlinear Dynamics and Chaos</em> by Strogatz, and I have some questions about how to make this concept rigorous. In this section the book considers equations of the form
<span class="math-container">$$ \hspace{4.5cm} \ddot{x} + x + \epsilon h(x,\dot{x... | Lutz Lehmann | 115,115 | <p>Already the simple perturbation series is a deliberate decomposition of the solution that gets its justification from its result, if it works. Of course one can make a theory of and for classes of perturbation problems where it works.</p>
<p>In the multiple time-scale expansion, the calculator gives themselves even ... |
2,220 | <p>Some of you (myself included) might remember how as a new user you struggle with finding stuff to answer, and hope to have these answers upvoted and accepted...</p>
<p>You really want that. You want to write comments, to the least but that requires 50 reputation.</p>
<p>As a result you look for old questions, poss... | Jeff Atwood | 153 | <p>The general guidance I give is twofold. Ask yourself …</p>
<ol>
<li><p>Could a student (of x skill level) learn anything useful/practical from this answer?</p></li>
<li><p>Would I be embarrassed to be associated with this answer?</p></li>
</ol>
<p>(Beyond that of course if it's a purely duplicate answer, it... |
3,212,499 | <p>I'm struggling to find a solution to this exercise:</p>
<blockquote>
<p>Consider a set of 65 girls and a set of 5 boys. Prove that there are 3
girls and 3 boys such that either every girl knows every boy or no
girl knows any of the boys.</p>
</blockquote>
<p>I know I should use the Ramsey Theorem but I have ... | Empy2 | 81,790 | <p>You only need 41 girls. Then either 21 girls each know 3 boys, or 21 each don't know 3 boys. There are ten trios of 3 boys, so one of the trios is known/not known to 3 girls.</p>
|
2,725,019 | <blockquote>
<p>Define $S := \{x ∈ \mathbb Q : x^2 ≤ 2\}$. Prove that $a:=\inf \{S\}$
satisfies $a^2 = 2$.</p>
</blockquote>
<p>Since a is a lower bound for S, we have $a^2\le 2$. if $a^2\neq 2$ then $a^2 < 2$ and we may set $\epsilon:= a^2 − 2 > 0$ and then I am not sure how to show it satisfies $a^2 = 2$.<... | felipeh | 73,723 | <p>It is possible to flesh out the reasoning in your question to solve a more general question. Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function, and consider the set
$$
S_f = \{x\in\mathbb{Q} \,:\, f(x) \leq 0\}.
$$
If $S_f$ is nonempty and $S_f\not=\mathbb{R}$, then $x^* = \inf S_f$ satisfies $f(x^*) = 0$. ... |
1,072,669 | <ul>
<li><p>Let <span class="math-container">$C_1,C_2,C_3,C_4$</span> be compact convexes of <span class="math-container">$\mathbb{R}^2$</span> such that <span class="math-container">$C_1\cap C_2\cap C_3\neq\emptyset,C_1\cap C_2\cap C_4\neq\emptyset,C_1\cap C_3\cap C_4\neq\emptyset,C_2\cap C_3\cap C_4\neq\emptyset$</sp... | Valerii Sokolov | 25,856 | <p>For second question, let $m=2, n=3$. We can easily draw tree circles each pair of which intersects and all three of them have empty intersection. So the answer is negative.</p>
<p>As for the first one, a convex set $X$ has, by definition, a property that for any two points $u, v \in X$ point $au + (1-a)v$ belongs t... |
3,669,269 | <p>I have learned that I can compute the moments of a random variable with this formula
<span class="math-container">$$\mu_n=\mbox{E}(X-\mbox{E}X)^n$$</span>
However, for the moment of order <span class="math-container">$1$</span> I can not use this, since I get <span class="math-container">$\mbox{E}(X-\mbox{E}X)=0$</s... | Community | -1 | <p>Indeed, the first order central moment is always zero and is not used.</p>
<p>Instead you can compute the first order <em>absolute</em> central moment,</p>
<p><span class="math-container">$$E(|X-E(X)|)$$</span> which is a measure of the spread, like the variance. It is not as efficient, but is more robust.</p>
|
3,278 | <h3>What are Community Promotion Ads?</h3>
<p>Community Promotion Ads are community-vetted advertisements that will show up on the main site, in the right sidebar. The purpose of this question is the vetting process. Images of the advertisements are provided, and community voting will enable the advertisements to be s... | John | 20,946 | <p><a href="http://www.proofwiki.org/wiki/Main_Page" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/BL9vf.png" alt="Alt Text"></a></p>
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3,278 | <h3>What are Community Promotion Ads?</h3>
<p>Community Promotion Ads are community-vetted advertisements that will show up on the main site, in the right sidebar. The purpose of this question is the vetting process. Images of the advertisements are provided, and community voting will enable the advertisements to be s... | Ilmari Karonen | 9,602 | <p><a href="https://oeis.org/search?q=0%2C1%2C3%2C6%2C2%2C7%2C13%2C20%2C12%2C21%2C11%2C22%2C10%2C23" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/JyF2j.png" alt="The On-Line Encyclopedia of Integer Sequences"></a></p>
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3,278 | <h3>What are Community Promotion Ads?</h3>
<p>Community Promotion Ads are community-vetted advertisements that will show up on the main site, in the right sidebar. The purpose of this question is the vetting process. Images of the advertisements are provided, and community voting will enable the advertisements to be s... | E.O. | 18,873 | <p><a href="http://ocw.mit.edu/courses/#mathematics" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/gTfnh.jpg" alt="MIT OpenCourseWare - Free Online Course Materials"></a></p>
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3,278 | <h3>What are Community Promotion Ads?</h3>
<p>Community Promotion Ads are community-vetted advertisements that will show up on the main site, in the right sidebar. The purpose of this question is the vetting process. Images of the advertisements are provided, and community voting will enable the advertisements to be s... | F'x | 3,406 | <p><a href="http://academia.stackexchange.com"><img src="https://i.stack.imgur.com/QlIlq.png" alt="Academia Stack Exchange"></a></p>
|
1,454,919 | <p>I am trying to understand derivative and I want to know intuitive and rigorous definitions for a curve and if derivative is lmited only to curves or not..</p>
| Surb | 154,545 | <p>A curve is the graph of an application \begin{align*}\gamma :A\subset \mathbb R&\longrightarrow \mathbb R^n\\ t&\mapsto (\alpha_1(t),...,\alpha_n(t))\end{align*}</p>
<p>where \begin{align*}
\alpha_i:A&\longrightarrow \mathbb R\\
t&\mapsto \alpha_i(t).
\end{align*}</p>
<p>And yes, derivative are lim... |
1,820,036 | <p>I'd be thankful if some could explain to me why the second equality is true...
I just can't figure it out. Maybe it's something really simple I am missing?</p>
<blockquote>
<p>$\displaystyle\lim_{\epsilon\to0}\frac{\det(Id+\epsilon H)-\det(Id)}{\epsilon}=\displaystyle\lim_{\epsilon\to0}\frac{1}{\epsilon}\left[\de... | Laurent.C | 346,533 | <p>actually, you're trying to calculate the differential $\phi$ of the function $\det : M_n(\mathbb{R}) \rightarrow \mathbb{R}$ at $I_n$, which is defined as
$$\forall H \in M_n(\mathbb{R}), \phi(H) = \lim\limits_{t \to 0} \dfrac{\det(I_n+tH)-\det(I_n)}{t}=
\dfrac{d}{dt}_{t=0} \det(I_n+tH)
$$</p>
<p>First, since $\det... |
2,639,013 | <p>Let $V$ a vector space over field $\mathbb{K}$ with inner product and let $U$ and $W$ subspaces of $V$ so that $U \subseteq W^{\perp}$ and $V = W+U$. Show that $U=W^{\perp}$.</p>
<p>I try this approach:
Let $w \in W+U$ so $w \in U$ and $\langle w,v \rangle = 0$ for all $v \in W$. In particular $\langle w,w \rangle ... | carmichael561 | 314,708 | <p>Note that $A=nI_n+vv^T$, where $v=(1,\dots,1)^T$. Using the <a href="https://en.wikipedia.org/wiki/Matrix_determinant_lemma" rel="nofollow noreferrer">matrix determinant lemma</a>, we have
$$ \det(nI_n+vv^T)=\Big(1+\frac{1}{n}v^Tv\Big)\det(nI_n)=2n^n$$</p>
|
2,639,013 | <p>Let $V$ a vector space over field $\mathbb{K}$ with inner product and let $U$ and $W$ subspaces of $V$ so that $U \subseteq W^{\perp}$ and $V = W+U$. Show that $U=W^{\perp}$.</p>
<p>I try this approach:
Let $w \in W+U$ so $w \in U$ and $\langle w,v \rangle = 0$ for all $v \in W$. In particular $\langle w,w \rangle ... | Joca Ramiro | 264,635 | <p>Writing $A=nI_n+U$, where $U$ is the matrix with all elements equal to one, the eigenvalues $\lambda$ are solutions to the characteristic polynomial $\det(U-(\lambda-n)I_n)=\det(U-\lambda'I_n)$, where $\lambda'=\lambda-n$. In this <a href="https://math.stackexchange.com/questions/217521/what-are-the-eigenvalues-of-m... |
244,115 | <p>I create a very large output</p>
<pre><code> D[x^100*E^(2*x^5)*Cos[x^2], {x, 137}]
</code></pre>
<p>I want to assign this to a function as</p>
<pre><code> f[x_]:= {very large output from previous command}
</code></pre>
<p>This allows me to evaluate that output for various values of <span class="math-container"... | Bob Hanlon | 9,362 | <pre><code>Clear["Global`*"]
f1[x_] = D[x^100*E^(2*x^5)*Cos[x^2], {x, 137}];
f2[x_] = D[x^100*E^(2*x^5)*Cos[x^2], {x, 137}] // Simplify;
</code></pre>
<p>While defining <code>f2</code> takes much longer,</p>
<pre><code>LeafCount /@ {f1[x], f2[x]}
(* {4274535, 6386} *)
AbsoluteTiming[#[3.]] & /@ {f1, f... |
2,495,918 | <p>A triangle is formed by the lines $x-2y-6=0$ , $ 3x−y+6=0$, $7x+4y−24=0$.</p>
<p>Find the equation of the line that bisects the inner angle of the triangle that is facing the side $7x+4y−24=0$.</p>
<p>I tried to find the intersect point of three equations by put them equal two by two. However, I don't know what to... | GAVD | 255,061 | <p>HINT: if point $P(x,y)$ is in the bisector of angle between two lines $x-2y-6=0$ and $3x-y+6=0$, then the distances from this point to each lines are equal. So, you get two lines which are bisector. Check which is the inner bisector.</p>
<p>HINT 2: See this <a href="https://en.wikipedia.org/wiki/Distance_from_a_poi... |
145,429 | <p>I have expression like this:</p>
<pre><code>expr = xuyz;
</code></pre>
<p>then</p>
<pre><code>Head[expr] = xuyz
</code></pre>
<p>But I wanted the product of four factors, so it should have been written as
<code>x*u*y*z</code> or <code>x u y z</code>, because Mathematica understands multiplication of four single... | Coolwater | 9,754 | <p>This returns <code>True</code> because of <code>xu</code> and/or <code>gh</code></p>
<pre><code>Max[StringLength[ToString /@ DeleteCases[Level[xu*y/gh, {-1}], _?NumericQ]]] > 1
</code></pre>
<blockquote>
<p>True</p>
</blockquote>
|
145,429 | <p>I have expression like this:</p>
<pre><code>expr = xuyz;
</code></pre>
<p>then</p>
<pre><code>Head[expr] = xuyz
</code></pre>
<p>But I wanted the product of four factors, so it should have been written as
<code>x*u*y*z</code> or <code>x u y z</code>, because Mathematica understands multiplication of four single... | m_goldberg | 3,066 | <p>I haven't thoroughly tested the following function, but at least it might be a good starting point for you to solve your problem.</p>
<pre><code>singleCharVarsQ[expr_] :=
AllTrue[
StringLength /@ SymbolName /@
Cases[expr, Except[_?NumericQ, _?AtomQ], ∞], # == 1 &]
singleCharVarsQ[x y z + x Sqrt[x... |
145,429 | <p>I have expression like this:</p>
<pre><code>expr = xuyz;
</code></pre>
<p>then</p>
<pre><code>Head[expr] = xuyz
</code></pre>
<p>But I wanted the product of four factors, so it should have been written as
<code>x*u*y*z</code> or <code>x u y z</code>, because Mathematica understands multiplication of four single... | kglr | 125 | <pre><code>f1 = Max@StringLength[SymbolName /@ Variables@#] == 1 &;
f1 /@ {xu*y, 3 x + w z^2}
</code></pre>
<blockquote>
<p>{False, True}</p>
</blockquote>
|
1,393,822 | <p>For a complex number $w$, or $a+bi$, is there a specific term for the value $w\overline{w}$, or $a^2+b^2$?</p>
| Barry Cipra | 86,747 | <p>$${n^{2.5}+(n+1)^{2.5}\over5}={n^{2.5}\over5}\left(1+\left(1+{1\over n}\right)^{2.5} \right)$$</p>
<p>and</p>
<p>$$\left(1+{1\over n}\right)^{2.5}\approx1+{2.5\over n}$$</p>
<p>so</p>
<p>$${n^{2.5}+(n+1)^{2.5}\over5}\approx{n^{2.5}\over5}\left(2+{2.5\over n}\right)={2\over5}n^{2.5}+{1\over2}n^{1.5}$$</p>
|
371,318 | <p>The original problem was to consider how many ways to make a wiring diagram out of $n$ resistors. When I thought about this I realized that if you can only connect in series and shunt. - Then this is the same as dividing an area with $n-1$ horizontal and vertical lines. When each line only divides one of the current... | Don Mintz | 151,541 |
<p>For $x$ > 0, define an
infinite number by the divergent geometric series:
$\displaystyle\sum_{i=0}^{n\rightarrow\infty} \left(\frac{x+1}{x}\right)^i $ </p>
<p>and define an infinitesimal number as the difference between a convergent geometric series and its sum:</p>
<p>$ x+1 -\displaystyle\sum_{i=0}^{n... |
498,694 | <p>So, I'm learning limits right now in calculus class.</p>
<p>When $x$ approaches infinity, what does this expression approach?</p>
<p>$$\frac{(x^x)}{(x!)}$$</p>
<p>Why? Since, the bottom is $x!$, doesn't it mean that the bottom goes to zero faster, therefore the whole thing approaches 0?</p>
| Vaidyanathan | 86,594 | <p>You may use Stirlings approximation to do this.</p>
<p>Let $$y = x^x/x! $$, then $$ln(y) = ln((x^x/x!)) = ln(x^x) - ln(x!)$$</p>
<p>The last term on the right hand side can be expanded as $$ ln(x!) = xln(x) - x $$ for large n. As a side note, this is a typical expansion used in physics.</p>
<p>Substituting back a... |
498,694 | <p>So, I'm learning limits right now in calculus class.</p>
<p>When $x$ approaches infinity, what does this expression approach?</p>
<p>$$\frac{(x^x)}{(x!)}$$</p>
<p>Why? Since, the bottom is $x!$, doesn't it mean that the bottom goes to zero faster, therefore the whole thing approaches 0?</p>
| N. S. | 9,176 | <p><strong>Hint</strong>
$$\frac{x^x}{x!} > \frac{x}{1}$$</p>
|
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